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A CHARACTERIZATION OF TOTALLY REAL CARLEMAN SETS AND AN APPLICATION

TO PRODUCTS OF STRATIFIED TOTALLY REAL SETS

BENEDIKT S. MAGNUSSON and ERLEND FORNÆSS WOLD

Abstract

We give a characterization of stratified totally real sets that admit Carleman approximation by entire functions. As an application we show that the product of two stratified totally real Carleman sets is a Carleman set.

1. Introduction

In one complex variable, the so called Carleman sets are well understood: A closed subsetXofCis a Carleman set if and only if (i) it has no interior, (ii)X is polynomially convex, and (iii)C\Xis locally connected at infinity (Keldych and Lavrentieff [1]). In the complex plane property (iii) is equivalent to what we call having bounded E-hulls, but inCnthere is no topological characterization.

InCn it is clear that (i)–(iii) is not sufficient for any type of approximation, as is shown by considering an affine complex line. In particular (i) must be substituted by some other “lack of complex structure”. In this article we prove the analogue of the one dimensional result for totally real sets.

Theorem1.1.LetMCn be a closed stratified totally real set. ThenM is a Carleman set if and only ifM is polynomially convex and has bounded E-hulls.

(For the definition of Carleman approximation and bounded E-hulls, see Section 2, and for the definition of a stratified totally real set, see Section 3.)

It is already known [2] that in the totally real setting, the property of bounded E-hulls implies Carleman approximation. The remaining result is therefore the following, which does not rely on the notion of being totally real:

Theorem1.2.IfMCn is a closed set which admits Carleman approx- imation by entire functions, thenMhas bounded E-hulls.

Received 25 October 2013.

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Theorem 1.2 generalizes the main theorem of [2] where the implication was shown under the assumption thatM is totally real and admitsC1-Carleman approximation. As an application of this theorem we prove the following partial answer to a question raised by E. L. Stout (private communication):

Theorem1.3.LetMjCnjbe stratified totally real sets forj =1,2which admit Carleman approximation by entire functions. ThenM1×M2Cn1×Cn2 admits Carleman approximation by entire functions.

(For the definition of a stratified totally real set see Section 3.)

The precise question is more general:ifMj are Carleman sets inCnj for j =1,2,isM1×M2Carleman?

A natural generalization of one of the main results of [2] which we will use in the proof of Theorem 1.3 is the following:

Theorem1.4.LetMCnbe a stratified totally real set which has bounded E-hulls, and letKCn be a compact set such thatKM is polynomially convex. Then anyfC(KM)O(K)is approximable in the Whitney C0-topology by entire functions.

As a corollary to Theorem 1.1 we also obtain the following:

Corollary1.5.Let MCn be a totally real manifold of classCk and assume thatM admits Carleman approximation by entire functions. ThenM admitsCk-Carleman approximation by entire functions.

Proof. By [2] this follows from the fact thatM has bounded E-hulls.

For more on the topic of Carleman approximation, seee.g.the monograph [5].

2. Proof of Theorem 1.2

Definition2.1. LetMCnbe a closed set. We say thatM is aCarleman set, or thatMadmits Carleman approximation by entire functions, ifO(Cn)is dense inC(M)in the WhitneyC0-topology.

Definition2.2. LetXCnbe a closed subset. Given a compact normal exhaustionXj of X we define the polynomial hull of X, denoted by X, by

X:= ∪jXj(this is independent of the exhaustion). We also seth(X):=X\X.

Ifh(X)is empty we say thatXis polynomially convex. Note thatXis closed and polynomially convex.

Definition2.3. We say that a closed setMCnhasbounded E-hullsif for any compact setKCnthe seth(KM)is bounded.

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We give two lemmas preparing for the proof of Theorem 1.2. The first one is a simple well known result à la Mittag-Leffler and Weierstrass, which we state for the lack of a suitable reference.

Lemma2.4.Let E = {xj}jN be a discrete sequence in Cn. Then for any sequence{aj}jNCthere exists an entire functionfO(Cn)withf (xj)= aj for alljN, and there exist holomorphic functionsf1, . . . , fnsuch that E=Z(f1, . . . , fn).

Proof. By Theorem 3.7 in [3] there exists an injective holomorphic map F = (f˜1, . . . ,f˜n):CnCn such that F (xj) = j · e1. For the first claim we letgO(C)be an entire function withg(j) = aj for all jNand set f = g◦ ˜f1. For the second claim letgO(C)be an entire function whose zero set is precisely{j}jN. Now setf1=g◦ ˜f1andfk = ˜fkfork=2, . . . , n.

Lemma 2.5.Let MCn be a Carleman set and letE = {xj}jN be a discrete set of points withECn\M. ThenMEis a Carleman set, with interpolation onE.

Proof. Assume thatq:MECandε:MER+are continuous functions. By Lemma 2.4 there exist functionsf1, . . . , fnO(Cn)such that fj(xk)=0 for allxkEandj =1, . . . , n, and such thatZ(f1, . . . , fn)M =

∅. So there exist continuous functionsgjC(M)such that g1·f1+ · · · +gn·fn =1

onM. SinceMadmits Carleman approximation we may approximate thegj’s by entire functionsg˜jO(Cn)such that the function

ϕ= ˜g1·f1+ · · · + ˜gn·fn

satisfiesϕ(x) =0 for allxM. Obviouslyϕ(z)=0 for allzE.

By the Mittag-Leffler Theorem there exists an entire functionhO(Cn) such thath(z)= q(z)for allzE. LetψC(M)be the functionψ(x):=

h(x)q(x)

ϕ(x) . SinceM admits Carleman approximation we may approximateψ by en entire functionσO(Cn), and if the approximation is good enough, the functionh(z)ϕ(z)·σ (z)isε-close toqonME.

Proof of Theorem 1.2. Aiming for a contradiction we assume thatM does not have bounded exhaustion hulls,i.e., there exists a compact setKsuch thath(KM)is not bounded. This implies that there is a discrete sequence of pointsE= {xj} ⊂ h(KM). By Lemma 2.5 there exists an entire function qO(Cn)such that|q(z)|< 12, forzM andq(xj)=j forxjE. Define C = qK. Forj > C we then have that|q(xj)|> supzKM{|q(z)|}which contradicts the assumption thatEh(KM).

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3. Proof of Theorem 1.4

Definition3.1. Let MCn be a closed set. We say thatM is astratified totally real set ifM is the increasing unionM0M1 ⊂ · · · ⊂ MN = M of closed sets, such thatMj \Mj1is a totally real set (a set which is locally contained in a totally real manifold) forj = 1, . . . , N, and withM0totally real.

The proof of Theorem 1.4 is an inductive construction depending on the following lemma [4, Theorem 4.5]

Lemma 3.2. Let KCn be a compact set, letMCn be a compact stratified totally real set, and assume thatKMis polynomially convex. Then any functionfC(KM)O(K)is uniformly approximable by entire functions.

Proof. Set Xj := KMj forj = 0, . . . , N. ThenX0 is polynomially convex (see the proof of Theorem 4.5 in [4]) and so it follows from [2] thatf|X0

is uniformly approximable by entire functions. The result is now immediate from Theorem 4.5 in [4].

Proof of Theorem1.4. Choose a normal exhaustionKj ofCn such that KjM is polynomially convex for eachj. Assume that we are givenfC(KM)O(K),εC(KM)withε(x) >0 for allxKM. Set K0 = K1 = K,f0 = f1 = f. We will construct an approximation off by induction onj, and we assume that, fork = 1, . . . , j, we have constructed fkC(KkM)O(Kk)such that

(1k) |fk(x)f (x)|< ε(x)/2 for all xM, and

(2k) fkfk1Kk−1 < (1/2)k.

ChooseχjC0(Kj+2)withχj ≡1 nearKj+1. For any 0< δj < (1/2)j+1 it follows from Lemma 3.2 that there exists an entire function gj such that

|gj(x)fj(x)|< δj for allxKj(MKj+2).

We definefj+1:=χj·gj+(1χj)(fj)onM, andfj+1:=gjnearKk+1. It is clear that we get

(2j+1) fj+1fjKj < δj < (1/2)j+1, and onM\Kj we have that

fj+1f =(fjf )+χj·(gjfj),

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so ifδj is sufficiently small we also get that

(1j+1) |fj+1(x)f (x)|< ε(x)/2 for all xM.

It follows from(2k)thatfk converges to an entire functionf˜, and it follows from(1k)that| ˜f (x)f (x)|< ε(x)for allxKM.

4. Proof of Theorem 1.3

Note first thatM1×M2is a stratified totally real set which is polynomially convex, so by Theorem 1.4 it suffices to show thatM1×M2 has bounded E-hulls. LetKCn1×Cn2be compact. Since bothM1andM2are Carleman sets, they have bounded exhaustion hulls by Theorem 1.2. Choose compact setsK˜j inCnj forj =1,2,withK ⊂ ˜K1× ˜K2. SetKj := ˜Kjh(K˜jMj) which are now compact sets.

We claim that(K1×K2)(M1×M2)is polynomially convex, from which it follows thath(K(M1×M2))K1×K2. Let(z0, w0)(Cn1 ×Cn2)\ [(K1×K2)(M1×M2)]. We consider several cases.

(i) z0/ K1M1. Here,K1M1is polynomially convex and we simply use a function in the variablez0only.

(ii) w0/K2M2. Analogous to (i).

(iii) z0M1K1. Thenw0/K2M2, so we are in case (ii).

(iv) z0M1 \K1. Then w0K2\M2, unless we are in case (ii). By Lemma 2.5 there existsfO(Cn2)such thatf (w0)=1 and|f (w)|<

1/2 for all wM2. Set N = fK2. By Theorem 1.4 there exists gO(Cn1)such thatgK1 <1/(2N),|g(z)|<3/2 for allzM1and g(z0)=1. Seth(z, w)=f (w)·g(z). Thenh(z0, w0)=1. For(z, w)K1×K2we have|h(z, w)| = |f (w)||g(z)| ≤ N ·1/(2N) = 1/2. If (z, w)M1×M2then|h(z, w)| = |f (w)||g(z)| ≤1/2·3/2=3/4.

(v) z0K1\M1. Thenw0M2\K2unless we are in case (ii), but this is the same as (iv) with the roles ofz0andw0switched.

REFERENCES

1. Keldych, M., and Lavrentieff, M.,Sur un problème de M. Carleman, C. R. (Doklady) Acad.

Sci. URSS (N. S.) 23 (1939), 746–748.

2. Manne, P. E., Wold, E. F., and Øvrelid, N.,Holomorphic convexity and Carleman approxim- ation by entire functions on Stein manifolds, Math. Ann. 351 (2011), no. 3, 571–585.

3. Rosay, J.-P., and Rudin, W.,Holomorphic maps fromCntoCn, Trans. Amer. Math. Soc. 310 (1988), no. 1, 47–86.

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4. Samuelsson, H., and Wold, E. F.,Uniform algebras and approximation on manifolds, Invent.

Math. 188 (2012), no. 3, 505–523.

5. Stout, E. L.,Polynomial convexity, Progress in Mathematics, vol. 261, Birkhäuser Boston, Inc., Boston, MA, 2007.

SCIENCE INSTITUTE UNIVERSITY OF ICELAND DUNHAGA 5

107 REYKJAVIK ICELAND

E-mail:bsm@hi.is

DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO P.O. BOX 1053 BLINDERN NO-0316 OSLO NORWAY

E-mail:erlendfw@math.uio.no

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